Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #388397 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14519 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,6,12,7,11)(3,13,15,24,17,28,16,8)(4,18,20,26,21,9,22,27)(14,25,30,23,29,31,19,32);(1,3)(2,8)(4,19)(5,15)(6,17)(7,16)(9,25)(10,13)(11,28)(12,24)(14,20)(18,31)(21,30)(22,29)(23,27)(26,32);(1,4,6,21)(2,9)(3,14)(5,22,7,20)(8,23,24,32)(10,26)(11,27)(12,18)(13,25,28,31)(15,19)(16,30)(17,29);(2,10)(3,15)(5,7)(9,26)(11,12)(13,28)(14,19)(16,17)(18,27)(20,22)(25,31)(29,30)Polycyclic presentation:
2^9|a^2=h,c^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72237065, #72237073, #72237090, … |
| maximal normal subgroup of | #72237065, #72237073, #72237090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 295595);